finding remaining trigonometric functions of theta

kmanson

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Dec 1, 2013
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Hi! I am confused about this. Additionally, we cannot use a calculator.
Here is the problem:
cos(theta)=-1/3, (pi/2)<theta<pi
So I know how to find tan, cot, csc, and sec, if I could only find out sin(theta). I also know that cos(pi/2)=cos(90)=0, and that cos(pi)=cos(180)=-1.
Does this possibly mean that, since -1/3 is a third of the difference between -1 and 0, that theta is (90+30), because 30 is a third of the difference between 180 and 90? If so, how would you find sin(120)?
 
Hi! I am confused about this. Additionally, we cannot use a calculator.
Here is the problem:
cos(theta)=-1/3, (pi/2)<theta<pi

Here \(\displaystyle x=-1~\&~r=3 \) so \(\displaystyle y=\sqrt 8 \)

\(\displaystyle \sin(\theta)=\dfrac{y}{r}~\&~\tan(\theta)=\dfrac{y}{x}\)

Then you have to find the co-functions.
 
Hi! I am confused about this. Additionally, we cannot use a calculator.
Here is the problem:
cos(theta)=-1/3, (pi/2)<theta<pi
So I know how to find tan, cot, csc, and sec, if I could only find out sin(theta). I also know that cos(pi/2)=cos(90)=0, and that cos(pi)=cos(180)=-1.
Does this possibly mean that, since -1/3 is a third of the difference between -1 and 0, that theta is (90+30), because 30 is a third of the difference between 180 and 90? If so, how would you find sin(120)?

what does cos2(theta) + sin2(theta) equal ? So what is sin(theta) in terms of cos(theta)?
 
Kmanson,

Draw a triangle to see what's up.

Cos is adjacent of hypotenuse.

so your adjacent will be -1 and your hypotenuse will be 3.

From pythagorean theorem, you can find the length of the side opposite theta: sqrt{32 - (-1)2} = sqrt{8} = 2sqrt{2}

So the side opposite angle theta is 2 sqrt{2}.
The adjacent side is -1 and the hypotenuse is 3.
Now, you can easily find any function you want, including sin.
sin = opposite / hypotenuse = 2 sqrt{2} / 3.

Hope that helps,
Amadeus
 
what does cos2(theta) + sin2(theta) equal ? So what is sin(theta) in terms of cos(theta)?

cos2(x) + sin2(x) = 1 for any angle.

right_triangle.gif'

From pythagorean theorem, you know that a2 + b2 = c2
If you divide both sides by c2, you get:

a2 / c2 + b2 / c2 = c2 / c2
a2 / c2 + b2 / c2 = 1

But a / c is basically sine theta, and b / c is cosine.
So we get sin2(theta) + cos2(theta) = 1.


Hope that helps,
Amadeus
 
cos2(x) + sin2(x) = 1 for any angle.

View attachment 3471'

From pythagorean theorem, you know that a2 + b2 = c2
If you divide both sides by c2, you get:

a2 / c2 + b2 / c2 = c2 / c2
a2 / c2 + b2 / c2 = 1

But a / c is basically sine theta, and b / c is cosine.
So we get sin2(theta) + cos2(theta) = 1.


Hope that helps,
Amadeus

I wasn't asking the question for myself. I was trying to guide the original poster towards an answer. It's a fairly common thing we do here.
 
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